Mixing-angle sensitivity of inclusive $\Xi_{bc}$ and $\Omega_{bc}$ lifetimes

Technical Summary: Mixing-angle sensitivity of inclusive $\Xi_{bc}$ and $\Omega_{bc}$ lifetimes

Problem Statement
The inclusive lifetimes of doubly heavy baryons containing a bottom and a charm quark ($bcq$) remain a theoretical challenge due to the unique spin composition of the heavy pair. Unlike systems with identical heavy quarks (e.g., $ccq$ or $bbq$), the non-identical $bc$ pair allows for both spin-zero ($S_{bc}=0$) and spin-one ($S_{bc}=1$) configurations. Hyperfine interactions mix these two basis states, forming physical states defined by a mixing angle $\phi_B$. Previous lifetime analyses have largely treated the $S_{bc}=0$ and $S_{bc}=1$ cases as separate, pure configurations. However, the physical states are superpositions, and the interference between these configurations via off-diagonal four-quark matrix elements ($T_{01}$) introduces a dependence on the mixing angle that cannot be captured by examining the pure limits alone. This paper addresses the sensitivity of inclusive lifetimes to this mixing angle for the $\Xi_{bc}$ and $\Omega_{bc}$ baryons.

Methodology
The authors employ the Heavy Quark Expansion (HQE) to separate short-distance coefficients from local hadronic matrix elements. The transition operator is expanded up to dimension-seven, incorporating:

• Partonic contributions: Calculated at Next-to-Leading Order (NLO) for chromomagnetic and semileptonic terms, and Next-to-Next-to-Leading Order (NNLO) for the partonic $C_3$ coefficients.
• Spectator effects: Dimension-six four-quark operators (W-exchange and Pauli interference) and dimension-seven corrections.
• Mixing Formalism: The physical baryon state is defined as $|B_{bc}(\phi_B)\rangle = \cos\phi_B |S_{bc}=1\rangle + \sin\phi_B |S_{bc}=0\rangle$. The expectation value of the transition operator $T$ is decomposed into diagonal terms ($T_{00}, T_{11}$) and an interference term proportional to $\sin(2\phi_B)\text{Re}(T_{01})$.

To evaluate the necessary hadronic matrix elements, the authors utilize two structurally distinct non-perturbative models to test the robustness of the results against spatial wave function variations:

1. Homogeneous Bag Model (HBM): Utilizes relativistic projected bag states with a static spherical bag radius.
2. Nonrelativistic Quark Model (NRQM): Uses Gaussian wave functions with Jacobi coordinates.

Both models calculate the spin-flavor overlap coefficients and the reduced spatial matrix elements for the four-quark operators. The analysis focuses on benchmark mixing angles derived from bag model spectra: $\phi_\Xi = -23.0^\circ$ and $\phi_\Omega = -23.5^\circ$.

Key Contributions

• Angular Dependence Analysis: The paper demonstrates that the inclusive lifetime is not a stationary function at the pure spin limits ($S_{bc}=0$ or $S_{bc}=1$). Instead, the extrema (minimum and maximum lifetimes) occur at mixed configurations with angles $|\phi_B| \simeq 27^\circ\text{--}63^\circ$. Consequently, predictions based solely on pure-spin assignments are insufficient to characterize the angular dependence.
• Model Independence: By comparing HBM and NRQM, the authors show that while the absolute baseline lifetimes and matrix element magnitudes differ significantly between models, the normalized angular response patterns are qualitatively identical. This suggests the mixing-angle sensitivity is a robust feature of the HQE framework rather than an artifact of a specific spatial wave function.
• Decomposition of Widths: The study provides a detailed breakdown of the decay widths, isolating the contributions from bottom decay ($\Gamma_b$), charm decay ($\Gamma_c$), and heavy-heavy spectator effects ($\Gamma_{bc}^{spec}$).

Results
At the benchmark mixing angles, the predicted inclusive lifetimes (in picoseconds) are:

• HBM:
  • $\tau(\Xi_{bc}^+) = 0.421^{+0.093}_{-0.093}$
  • $\tau(\Xi_{bc}^0) = 0.075^{+0.035}_{-0.010}$
  • $\tau(\Omega_{bc}^0) = 0.223^{+0.074}_{-0.043}$
• NRQM:
  • $\tau(\Xi_{bc}^+) = 0.367^{+0.076}_{-0.070}$
  • $\tau(\Xi_{bc}^0) = 0.100^{+0.032}_{-0.014}$
  • $\tau(\Omega_{bc}^0) = 0.229^{+0.066}_{-0.042}$

Physical Interpretation of Hierarchy:
The lifetime hierarchy $\tau(\Xi_{bc}^+) > \tau(\Omega_{bc}^0) > \tau(\Xi_{bc}^0)$ is driven primarily by charm-initiated spectator effects:

1. $\Xi_{bc}^+$: Experiences destructive Pauli interference ($cu$ channel), suppressing the decay width and resulting in the longest lifetime.
2. $\Xi_{bc}^0$: Experiences constructive W-exchange ($cd$ channel), significantly enhancing the decay width and resulting in the shortest lifetime.
3. $\Omega_{bc}^0$: Experiences constructive Pauli interference ($cs$ channel) at dimension six, which is partially cancelled by dimension-seven corrections, placing its lifetime between the other two.

Relative to the unmixed $S_{bc}=1$ reference points, the mixing reduces the lifetimes of $\Xi_{bc}^0$ and $\Omega_{bc}^0$ by approximately 17%–19%, while increasing the lifetime of $\Xi_{bc}^+$ by 5%–10%.

Significance
The paper claims that the inclusion of mixing-angle dependence is essential for accurate lifetime predictions of $bc$ baryons. The finding that extrema do not lie at pure spin configurations implies that theoretical uncertainties in mixing angles translate directly into significant uncertainties in lifetime predictions. Furthermore, the consistency of the angular response across the HBM and NRQM models strengthens the reliability of the HQE prediction for the mixing sensitivity, independent of the specific choice of hadronic wave function. These results are intended to guide the interpretation of future experimental signals and the design of search strategies for $\Xi_{bc}$ and $\Omega_{bc}$ states at facilities like LHCb.